Why is a sample statistic a random variable?
Because it's not defined in advance
The definition of random variable is a variable that doesn't have a value until some experiment is performed, as opposed to a variable in an algebra equation, which has a fixed value to be solved for.
Quick example: Roll a 6-sided die and let X = the number that comes up. X could have any value from 1 - 6 but we won't know until the die is rolled. It's not a number that can be "solved" for as in Algebra.
It's not that the sample statistic is itself a random variable. A statistic is a number that describes a set of data. Each data point is the value of a random variable for a different experiment.
Since your question is in the "sampling distribution of a proportion" category, consider the variable X = 1 if you voted for Candidate Smith and X=0 if you voted for candidate Jones.
If you're Smith's campaign manager and 180/300 polled chose Smith, it means that 300 experiments had X=1 180 times and X=0 120 times. So X is the random variable. The sample statistic is